Model Theory by Chang and Keisler references $R(\omega)$ frequently, usually in the context of models $\langle R(\omega), \in\rangle$ of ZF.

What does this notation mean, specifically? From the context, it looks like it's denoting the "usual" sets of ZF, perhaps more formally defined as "All sets reachable by repeated subset/powerset operations with from $\omega$". However, the book never explicitly defines it (or if it does, I missed it).

What's the formal definition of this notation? More importantly, where can I find notation that authors think is common knowledge next time that I miss something while reading a textbook? It's not easy to google $R(\omega)$ and get pertinent results.

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  • $\begingroup$ Most books have an index of symbols at the back (my copy of Chang & Keisler does), which you can use to find where things are first defined. $R(\omega)$ is defined inductively on p. 45, and more clearly on p. 588. $\endgroup$ – user82196 Oct 28 '13 at 1:55
  • $\begingroup$ Ah, thanks. It helps to know that $R(\omega)$ is called the "rank function". $\endgroup$ – Nate Oct 28 '13 at 4:52

For each ordinal $\alpha$, $R(\alpha)$ is the $\alpha$-th stage of the cumulative von Neumann hierarchy, more often denoted by $V_\alpha$. In particular, $R(\omega)=V_\omega$ is the set of hereditarily finite sets, i.e., the finite sets $x$ such that every element of $x$ is finite, every element of an element of $x$ is finite, and so on.

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